On 02/04/2013 02:40 PM, Paul Elliott wrote:
> Suppose a physicist is using a mathematical theory saying
> that there are an uncountable number of points in space.
>> In view of Skolem's result, what would such a statement mean really?
Current physical theories are not meant to be perfect. The use of a
mathematical model does not mean that the model is an exact literal
description of the physical world, but only that it makes good
predictions. At this time, we do not know whether at Planck scale,
space is continuous or discrete (or even whether points in space are
physically meaningful at Planck scale). Even if space is found to be
discrete, many models will continue to use continuous space as a very
useful simplifying assumption. Using Skolem hulls would only complicate
model formalism.
The metaphysical Skolem hull question is best addressed using Occam's
razor: Simpler explanations are preferred to more complicated ones, and
it is simpler to claim that directed distance can be any real number as
opposed to say any real number definable in second order arithmetic.
Also, Skolem hulls are arbitrary because they depend on the
expressiveness of the language in which they are defined, and, in the
absence of uniformization, on the well-ordering.
> What experiment could convince us about the number of points in space being countable/uncountable?
An enumeration of points that is linked to experimental results by a
general theory would be empirical evidence for countability of space
points (assuming that using a different set of points would make the
theory contrived). Conversely, failure to find such enumeration would
be evidence of continuous space (especially when physical theories are
naturally described in terms of a continuum). Other evidence is
possible as well (including whether physical laws are recursive).
However, if true, uncountability of space is not directly testable by
finite observers, and ascertaining it uses philosophical reasoning.
Sincerely,
Dmytro Taranovsky